For the penalty parameter \(\beta \)  and denoising parameter \(\sigma \) , generally they should be tuned according to the noise level \(\delta \) . According to the Morozov's discrepancy principle [1]}, the value of \(\alpha \)  in (REF ) is positively correlated with the noise level \(\delta \) , and the parameters \(\sigma \)  in \(\mathcal {C}_{\sigma }\)  and \(\theta =\frac{\alpha }{\beta }\)  in (REF ) play the same role of controlling the rate of denoising. Hence, \(\sigma \)  should be proportional to \(\theta =\frac{\alpha }{\beta }\)  and \(\beta \sigma \)  should be positively correlated with the noise level \(\delta \) . In our numerical experiments, we tune the parameters \(\sigma \)  and \(\beta \)  such that \(\beta \sigma \)  is proportional to the noise level of the observation, i.e., \(\beta \sigma \sim \delta \) . In Table REF , we list the tuned values of \(\beta \)  and \(\sigma \)  for the cases where the noise levels are \(\delta =0.01\) , \(0.05\)  and \(0.1\) , respectively. These parameters are kept as constants for different finite element meshes.
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